Bayesian analysis considers population parameters
to be random, not fixed

Old information, or subjective judgment, is used to determine a prior
distribution for these population parameters

It makes a great deal of practical sense to
use all the information available, old and/or new, objective or subjective,
when making decisions under uncertainty. This is especially true when the
consequences of the decisions can have a significant impact, financial
or otherwise. Most of us make everyday personal decisions this way, using
an intuitive process based on our experience and subjective judgments.

Mainstream statistical analysis, however, seeks objectivity by generally
restricting the information used in an analysis to that obtained from a
current set of clearly relevant data. Prior knowledge is not used except
to suggest the choice of a particular population model to "fit" to the
data, and this choice is later checked against the data for reasonableness.

Lifetime or repair models, as we saw earlier when we looked at repairable
and non repairable reliability population models,
have one or more unknown parameters. The classical statistical approach
considers these parameters as fixed but unknown constants to be estimated
(i.e., "guessed at") using sample data taken randomly from the population
of interest. A confidence interval for an unknown parameter is really a
frequency statement about the likelihood that numbers calculated from a
sample capture the true parameter. Strictly speaking, one cannot make probability
statements about the true parameter since it is fixed, not random.

The Bayesian approach, on the other hand, treats these population
model parameters as random, not fixed, quantities. Before looking at the
current data, we use old information, or even subjective judgments, to
construct a prior distribution model for these parameters. This
model expresses our starting assessment about how likely various values
of the unknown parameters are. We then make use of the current data (via
Baye's
formula) to revise this starting assessment, deriving what is called
the posterior distribution model for the population model parameters.
Parameter estimates, along with confidence intervals (known as credibility
intervals), are calculated directly from the posterior distribution.
Credibility
intervals are legitimate probability statements about the unknown parameters,
since these parameters now are considered random, not fixed.

It is unlikely in most applications that data will ever exist to validate
a chosen prior distribution model. Parametric Bayesian prior models are
chosen because of their flexibility and mathematical convenience. In particular,
conjugate priors (defined below) are a natural and popular choice
of Bayesian prior distribution models.

Bayes formula provides the mathematical tool
that combines prior knowledge with current data to produce a posterior
distribution

Bayes formula is a useful equation from probability
theory that expresses the conditional probability of an event A occurring,
given that the event \(B\)
has occurred (written P\((A|B)\)),
in terms of
unconditional probabilities and the probability the event \(B\)
has occurred, given that \(A\)
has occurred. In other words, Bayes formula
inverts which of the events is the conditioning event. The formula is
$$ \mbox{P}(A|B) = \frac{\mbox{P}(A,B)}{\mbox{P}(B)} = \frac{\mbox{P}(A) \cdot \mbox{P}(B|A)}{\mbox{P}(B)} \, , $$
and P(\(B\))
in the denominator is further expanded by using the so-called
"Law of Total Probability" to write
$$ \mbox{P}(B) = \sum_{i=1}^n \mbox{P}(B|A_i) \mbox{ P}(A_i) \, , $$
with the events \(A_i\)
being mutually exclusive and exhausting
all possibilities and including the event \(A\)
as one of the \(A_i\).

The same formula, written in terms of probability density function models,
takes the form:
$$ g(\lambda | x) = \frac{f(x | \lambda) g(\lambda)}{\int_0^\infty f(x | \lambda) g(\lambda) d \lambda} \, , $$
where \(f(x | \lambda)\)
is the probability model, or likelihood function, for the observed data \(x\)
given the unknown parameter (or parameters) \(\lambda\), \(g(\lambda)\)
is the
prior distribution model for \(\lambda\),
and \(g(\lambda | x)\)
is the posterior distribution model for \(\lambda\)
given that the data \(x\)
have been observed.

When \(g(\lambda | x)\)
and \(g(\lambda)\)
both belong to the same distribution family, \(g(\lambda)\)
and \(f(x | \lambda)\)
are called
conjugate distributions and \(g(\lambda)\)
is the conjugate prior for \(f(x | \lambda)\).
For example, the Beta distribution model is a conjugate prior for the proportion
of successes \(p\)
when samples have a binomial distribution. And the
Gamma model is a conjugate prior for the failure rate \(\lambda\)
when sampling failure times or repair times from an exponentially distributed
population. This latter conjugate pair (gamma, exponential) is used extensively
in Bayesian system reliability applications.

How Bayes Methodology is used in System Reliability Evaluation

Bayesian system reliability evaluation assumes
the system MTBF is a random quantity "chosen" according to a prior distribution
model

Models and assumptions for using Bayes methodology
will be described in a later section.
Here we compare the classical paradigm versus the Bayesian paradigm when
system reliability follows the HPP or exponential
model (i.e., the flat portion of the Bathtub Curve).

Classical Paradigm For System Reliability Evaluation:

The MTBF is one fixed unknown value - there is no “probability” associated
with it

Failure data from a test or observation period allows you to make inferences
about the value of the true unknown MTBF

No other data are used and no “judgment” - the procedure is objective and
based solely on the test data and the assumed HPP model

Bayesian Paradigm For System Reliability Evaluation:

The MTBF is a random quantity with a probability distribution

The particular piece of equipment or system you are testing “chooses” an
MTBF from this distribution and you observe failure data that follow an
HPP model with that MTBF

Prior to running the test, you already have some idea of what the MTBF
probability distribution looks like based on prior test data or an consensus
engineering judgment

Advantages
and Disadvantages of using Bayes Methodology

Pro's and con's for using Bayesian methods

While the primary motivation to use Bayesian
reliability methods is typically a desire to save on test time and materials
cost, there are other factors that should also be taken into account. The
table below summarizes some of these "good news" and "bad news" considerations.

Bayesian Paradigm: Advantages and Disadvantages

Pro's

Con's

Uses prior information - this "makes sense"

If the prior information is encouraging, less new testing may be needed
to confirm a desired MTBF at a given confidence